Instead of a superposition of a several Maxwellians, let
us consider the simpler non-thermal distribution:

This generalized Lorentzian function is
very convenient to model observed velocity distributions
([Vasyliunas 1968]), since it is quasi-Maxwellian at low and thermal
energies, while its non-thermal tail decreases as a power-law
at high energies, as generally observed in space plasmas; this is in
line with the fact that particles of higher energy have larger
free paths, and are thus less likely to achieve partial
equilibrium. A generating process for such distributions has
been suggested recently ([Collier 1993]). For typical space plasmas,
generally lies in the range 2-6.

This ``Kappa" distribution tends to a Maxwellian for since

In this limit, all the temperatures .
For
finite , however, the temperatures are different and
increase with q. In particular the traditional temperature is

and the effective temperature is

The larger , the closer the distribution is to a Maxwellian,
and the closer the 's are to .

Substituting (10) into (4), one
sees that the
distribution at distance z is still a Kappa function having the
same . In addition, as a consequence of the form (10),
we have

Substituting this relationship into the integral (6) and
changing variables
to recover in the integrand, we get

(with , in order that the integrals converge). Since
the density is the moment of order q = 0, this yields

Thus , and, since
, we deduce

The 's and thus follow a polytrope law, which is independent
of q. This generalizes the result of Scudder (1992a) to all the
temperatures , and in particular to the temperature
given by our measurement.

Hence with a Kappa distribution, the density and temperature
obey a polytrope law, not only when the temperature is defined
from the mean particle energy, but also when it is based on
other moments of the distribution, a situation encountered with
some measuring techniques.